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Appl. Math. Mech. -Engl. Ed.
DOI 10.1007/s10483-014-1781-9
c
Shanghai University and Springer-Verlag
Berlin Heidelberg 2014
Applied Mathematics
and Mechanics
(English Edition)
Dual solutions in MHD stagnation-point flow of Prandtl fluid
impinging on shrinking sheet∗
N. S. AKBAR1, Z. H. KHAN2, R. U. HAQ3, S. NADEEM3
(1. Department of Basic Science & Humanities, College of Electrical & Mechanical Engineering
(CEME), National University of Sciences and Technology, Islamabad 46000, Pakistan;
2. School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China;
3. Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan)
Abstract The present article investigates the dual nature of the solution of the magneto-
hydrodynamic (MHD) stagnation-point flow of a Prandtl fluid model towards a shrinking
surface. The self-similar nonlinear ordinary differential equations are solved numerically
using the shooting method. It is found that the dual solutions of the flow exist for cer-
tain values of the velocity ratio parameter. The special case of the first branch solutions
(the classical Newtonian fluid model) is compared with the present numerical results of
stretching flow. The results are found to be in good agreement. It is also shown that the
boundary layer thickness for the second solution is thicker than that for the first solution.
Key words stagnation-point flow, shrinking sheet, Prandtl fluid, magnetohydrody-
namic (MHD), dual solution, shooting method
Chinese Library Classification O361.3
2010 Mathematics Subject Classification 76E25, 76W05
1 Introduction
A number of studies have been reported in the literature focusing on the stagnation-point
flow towards a stretching sheet because of its industrial and engineering applications such
as extrusion, paper production, insulating materials, glass drawing, and continuous casting.
Hiemenz[1] initiated the two-dimensional stagnation-point flow. He discovered that the Navier-
Stokes equations governing the flow can be transformed into an ordinary differential equation
(ODE) of the third-order by using the similarity transformation. The boundary layer flow
over a stretching surface was studied by Sakiadis[2]. He modeled the laminar boundary-layer
behavior on a moving continuous flat surface and presented the numerical solutions for the
boundary-layer equations. Later on, this idea was extended by Crane[3] for both linear and
exponentially stretching sheets. Chiam[4] discussed the steady two-dimensional stagnation-point
flow of a viscous fluid towards a stretching sheet. He made the analysis with the assumption
that the sheet was stretched in its own plane with a velocity proportional to the distance
from the stagnation point. Wang[5] studied the free convection on a vertical stretching surface.
Nazar et al.[6] analyzed the unsteady two-dimensional stagnation-point flow of an incompressible
∗Received Jul. 23, 2013/ Revised Dec. 27, 2013
Corresponding author Z. H. KHAN, E-mail: zafarhayyatkhan@gmail.com
2 N. S. AKBAR, Z. H. KHAN, R. U. HEQ, and S. NADEEM
viscous fluid over a flat deformable sheet. The magnetohydrodynamic (MHD) stagnation-point
flow towards a stretching vertical sheet was discussed by Ishak et al.[7]. The two-dimensional
stagnation-point flow of a viscoelastic fluids is studied by Sadeghy et al.[8] assuming that the
fluid obeyed the upper-convected Maxwell (UCM) model. The boundary-layer hypothesis is
used to simplify the equations of motion which are further reduced to a single nonlinear third-
order ordinary ODE with the idea of the stream function coupled with the technique of the
similarity solution. The resulting equations were solved using the Chebyshev pseudo-spectral
collocation-point method. Attia[9] made an analysis of the steady hydromagnetic laminar three-
dimensional stagnation-point flow of an incompressible viscous fluid impinging on a permeable
stretching surface with heat generation or absorption.
MHD flows play an important role in the motion of fluids. Normally, a uniform magnetic field
is applied normal to the plate which is maintained at a constant temperature. The steady MHD
mixed convection flow of a viscoelastic fluid in the neighborhood of two-dimensional stagnation-
points with a magnetic field has been investigated by Kumari and Nath[10] considering the UCM
model. The boundary layer theory can simplify the equations of motion, the induced magnetic
field, and the energy into three coupled nonlinear ODEs. These equations were finally solved
by using the finite difference method. The results indicated that the increase in the elasticity
number causes the reduction in the surface velocity gradient and the surface heat transfer.
The dual solutions of the boundary layer flow over moving surfaces are of practical im-
portance in engineering analysis. It gives the possibility to determine the most realistic and
physically meaningful solution. Considerable amount of researches have been carried out to
investigate the multiple solutions of the boundary layer flows driven by moving surfaces. Kemp
and Acrivos[11] found the dual solutions for a moving-wall boundary layer reverse flow. Riley
and Weidman[12] presented multiple solutions for the Falker-Skan equation with a stretching
boundary. Ingham[13] presented the dual solutions of a steady mixed convection boundary layer
assisting flow over a moving vertical flat plate. Ridha and Curie[14] found the dual solutions
for both assisting and opposing flows. Recently, Mahapatra et al.[15] showed that the upper
branch solution was always stable whereas the lower branch was unstable. Moreover, the stable
branch was physically meaningful. Makinde et al.[16] showed that dual solutions existed for
the stagnation-point flow of an electrically conducting nanofluid towards a vertically stretching
sheet.
In the present article, we discuss the stagnation-point flow of a Prandtl fluid towards a
shrinking sheet with the magnetic field. To the best of authors′knowledge, no such investigation
has been carried out yet. The main ob jective of the article is to discuss the dual solutions for
both stagnation points and shrinking flows.
2 Mathematical formulation
We consider a two-dimensional stagnation-point flow of an incompressible Prandtl fluid over
a wall coinciding with the plane y= 0. The flow is confined to the plane y > 0, and the sheet
shrinks linearly along the x-axis. Moreover, we consider the effects of stagnation away from the
x-axis, and the uniform magnetic field is applied normal to the fluid flow (see Fig.1).
The extra stress tensor for the Prandtl fluid is defined as[17]
τ=Aarc sin1
C∂¯u
∂¯y2+∂¯v
∂¯x21
2
∂¯u
∂¯y2+∂¯v
∂¯x21
2
∂¯u
∂¯y,(1)
where Aand Care material constants of the Prandtl fluid model. The flow equations for the
Dual solutions in MHD stagnation-point flow of Prandtl fluid impinging on shrinking sheet 3
Fig. 1 Flow configuration and coordinate system
Prandtl fluid model after applying the boundary layer approximations can be defined as follows:
∂u
∂x +∂v
∂y = 0,(2)
u∂u
∂x +v∂u
∂y =ue
∂ue
∂x +νA
C
∂2u
∂y2+νA
2C3∂u
∂y 2∂2u
∂y2−σB2
0
ρ(u−ue),(3)
where uand vare the velocity components along the x- and y-axes, νis the kinematic viscosity,
B0is the magnetic field, σis the permeability of the fluid, and ρis the density of the fluid.
The flow velocity outside the boundary layer (inviscid fluid) is ue(x) = ax, where a > 0. From
Eq. (3), we can see that most of the liquid metals possess low conductivity which may lead to
small electrical currents generated by the fluid flow in the presence of a magnetic field. In other
words, the magnetic Reynolds number is much smaller than one[18–19].
The corresponding boundary conditions are
u=uw(x) = bx, v =vw,at y= 0,
u→ax, as y→ ∞,(4)
where a > 0 is the constant, and uwis the velocity at wall.
Introduce the following similarity transformations:
η=ra
νy, u =axf′(η), v =−√aνf (η).(5)
Invoking transformations (5), Eqs. (2)−(4) take the forms
P rf ′′′ −(f′)2+ff ′′ +βf ′′′ (f′′ )2+Ha2(1 −f′) + 1 = 0,(6)
f=s, f′=λ=b/a, at η= 0,
f′→1 at η→ ∞.(7)
In the above equations, α=A
C,δ=a3x2A
2Cν are the Prandtl parameter, λ=b
ais the stretching
(λ > 0) or shrinking (λ < 0) parameter, M2=σB2
0
ρa is the magnetic field parameter, and sis
the mass transfer parameter with s > 0 for suction and s < 0 for injection.
After using the boundary layer approximations, the wall shear stress τwcan be given by
τw=A
C
∂u
∂y +A
6C3∂u
∂y 3
.(8)
4 N. S. AKBAR, Z. H. KHAN, R. U. HEQ, and S. NADEEM
The coefficient of the skin friction is defined by
Cf=τw
ρu2
w
.(9)
In the dimensionless form, the skin friction is defined as
Re
1
2
xCf= (P rf ′′(η) + β(f′′ (η))3)η=0 .(10)
3 Numerical metho d for solutions
Numerical solutions to the governing ordinary differential Eq.(6) subjected to the boundary
conditions (7) are obtained by a shooting method. First, we convert the boundary value problem
(BVP) into the initial value problem (IVP) and assume a suitable finite value for the far field
boundary condition, i.e., η→ ∞, saying η∞. Then, we set the following first-order system:
f′=p, f′′ =p′=q, f ′′′ =q′=p2−fq −H a2(1 −p)−1
P r +βq2(11)
with the boundary conditions
f(0) = s, f′(0) = p(0) = λ. (12)
To solve (11) with (12) as an IVP, the values for q(0), i.e., f′′(0), is needed. However, no such
values are given prior to the computation. The initial guess values of f′′(0) are chosen, and the
fourth-order Runge-Kutta method is applied to obtain a solution. We compare the calculated
values of f′(η) at the far field boundary condition η∞(= 20) with the given boundary condition
f′(η)→ ∞, and the values of f′′(0) are adjusted by the Secant method for better approximation.
The step-size is taken as ∆η= 0.01, and the accuracy to the fifth decimal place is regarded as
the criterion of convergence. It is important to note that the dual solutions are obtained by
setting two different initial guesses for the values of f′′(0), where both profiles (first and second
solutions) satisfy the far field boundary condition (7) asymptotically but with different shapes.
To validate the accuracy of the proposed numerical scheme, the obtained results correspond-
ing to the skin-friction coefficient f′′(0) (for the Newtonian fluid case, i.e., β= 0) are compared
with those available in the literatures[20–22] in Tables 1 and 2 for the stretching case (λ > 0)
and the shrinking case (λ < 0).
Table 1 Comparison of values of coefficient of skin-friction (with Ha = 0) for shrinking sheet with
different values of λ
λ
Present results
α= 1, β= 0 Mahapatra and Nandy[22] Wang[21] Lok et al.[20]
0.0 1.232 6 1.232 6 1.232 6 −
0.1 1.146 6 1.146 6 1.146 6 −
0.2 1.051 1 1.051 1 1.051 1 −
0.5 0.713 3 0.713 3 0.713 3 0.713 3
1.0 0.000 0 0.000 0 0.000 0 −
2.0 −1.887 3 −1.887 3 −1.887 3 −1.887 3
5.0 −10.264 7 −10.264 7 −10.264 8 −10.264 8
Dual solutions in MHD stagnation-point flow of Prandtl fluid impinging on shrinking sheet 5
Table 2 Comparison of values of coefficient of skin-friction (with Ha = 0) for shrinking sheet with
different values of λ
λPresent results for P r=1, β=0 Results of Ref. [22]
First solution Second solution First solution Second solution
−0.25 1.402 2 −1.402 2 −
−0.50 1.495 6 −1.495 7 −
−0.75 1.489 3 −1.489 3 −
−1.0 1.328 8 0.000 0 1.328 8 0.000 0
−1.10 1.186 7 0.049 2 1.186 7 0.049 2
−1.15 1.082 2 0.116 7 1.082 2 0.116 7
−1.20 0.932 5 0.233 6 0.932 4 0.233 6
−1.246 0.554 3 0.554 2 0.584 4 0.554 2
4 Results and discussion
In the present section, we discuss the effects of the physical parameters such as the suc-
tion/injection parameter s, the Prandtl parameter α, the elastic parameter β, and the magnetic
parameter Mon both the velocity profile and the skin friction coefficient. The effects of the
various fluid flow parameters on the skin friction coefficient are presented in Figs.2−4. From
Fig. 2, it can be observed that for fixed suction and MHD parameters, the local skin friction
increases as the elastic parameter (β) decreases. Meanwhile, the existence of the dual solution
for the shrinking sheet case is found when λc< λ < 0, where the critical values of λare found
to be λc≈ −4.69, −4.78, and −5.47 for β= 0.3, 0.2, and 0.1, respectively. Moreover, as
the elastic parameter increases, the curve for the local skin friction firstly decreases gradually,
and then decreases to zero at λc=−4. It is also observed that for decreasing values of the
elastic parameter β, both the branches of the skin friction coefficient present the same sort of
increasing behaviors.
Fig. 2 Variations of skin friction with λfor
different values of β
Fig. 3 Variation of skin friction with λfor dif-
ferent values of M
It is clear from Figs. 2−4 that dual solutions exist beyond the critical value (turning point)
λc. Large imposition of suction is required so that dual solutions are possible for the flow with
large magnetic parameters. In reality, between these two solutions, only one solution is stable.
The first solution is assumed to be physically stable because its solution is the continuation of
the case of injection (s < 0). The second has negative values of the skin friction coefficient.
This solution shows the occurrence of flow separation and reverse, which cause the difficulties
in the numerical computation. Merkin[23] , Weidman et al.[24], Paullet and Weidman[25], and
Harris et al.[26] have presented the mathematical proof of the conjecture of dual numerical
6 N. S. AKBAR, Z. H. KHAN, R. U. HEQ, and S. NADEEM
solutions. They performed stability analyses and revealed that the solutions along the upper
branch (first solution) are linearly stable while those on the lower branch (second solutions) are
linearly unstable.
The effects of the Hartmann number Mand the mass transfer parameter sare presented
in Figs. 3 and 4, respectively, while the rest of the parameters are fixed. It is observed from
Fig. 3 that in the present region for the values of M, we can have dual solutions of the skin
friction coefficient against λ. Figure 3 shows that in both the cases, lower and upper branches
give the same increasing behavior for higher values of M. It is observed from Fig. 4 that both
curves have the same increasing behavior for higher values of s. Finally, we show that for the
stretching/shrinking parameter λ, dual solutions exist.
Fig. 4 Variation of skin friction with λfor dif-
ferent values of s
Fig. 5 Velocity distribution for different val-
ues of α
The variation of the velocity profile f′(η) with various fluid flow parameters are plotted in
Figs. 5−7. The effects of the Prandtl fluid parameter P r and the elastic parameter βare plotted
in Figs. 5−6. The dual velocity profiles show that the velocity decreases with the increases in α
and βwhile conversely increases for the second solution. It is also noted that the boundary layer
is thicker for the second solution in comparison with that for the first solution. The effects of
the suction/injection parameter son the dual velocity profiles are shown in Fig. 7. As the mass
transfer parameter sincreases in the magnitude, both the profiles increase. For large values of
the far field boundary η∞, the second solution velocity profiles decrease with the increase in s.
Fig. 6 Velocity distribution for different val-
ues of β
Fig. 7 Velocity distribution for different val-
ues of s
Dual solutions in MHD stagnation-point flow of Prandtl fluid impinging on shrinking sheet 7
5 Conclusions
We theoretically study the boundary layer stagnation-point flow of the Prandtl fluid model
towards a shrinking sheet in the presence of a magnetic field. We discuss the effects of the
suction/injection parameter s, the Prandtl parameter α, the elastic parameter β, and the
magnetic parameter Mon the fluid flow. The key conclusions of this study are as follows.
(i) We show that dual solutions exist for the proposed Prandtl fluid flow model.
(ii) The skin friction increases with the increases in the suction/injection parameter sand
the magnetic parameter M, whereas with the increase in the elastic parameter β, the skin
friction decreases.
(iii) It has been observed that the boundary layer thickness for the second solution is thicker
than that for the first solution.
(iv) For increasing suction/injection parameter s, the boundary layer thickness for both
solutions decreases.
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